A manufacturing company monitors the downtimes on its key fabricating machines. Data for the downtimes on one particular machine are given in the accompanying data table. Assuming that the downtimes are exponentially​ distributed, use the data to find the probability that a specific downtime will exceed

0.90.9

minuteminute.

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Part 1

The probability that a specific downtime will exceed

0.90.9

minuteminute

is

enter your response here.

To find the probability that a specific downtime will exceed \( 0.9 \) minutes, assuming that the downtimes follow an **exponential distribution**, we can use the **exponential probability density function** (PDF). The formula for the probability that a downtime \( X \) exceeds a certain value \( x \) is given by:

\[
P(X > x) = e^{-\lambda x}
\]

Where:
- \( \lambda \) is the rate parameter of the exponential distribution.
- \( x \) is the specified downtime, in this case, \( x = 0.9 \) minutes.

### Step 1: Estimate \( \lambda \)
The rate parameter \( \lambda \) can be estimated as the reciprocal of the **mean** of the observed downtimes. To proceed, we would need the data table to calculate the mean downtime and find \( \lambda \).

If you provide the data, I can help you calculate the mean downtime and complete the rest of the solution.

Would you like to upload the data table for further analysis?

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Here are some related questions you might find useful:

1. How do we calculate the mean for an exponentially distributed variable?
2. What is the relationship between the rate parameter \( \lambda \) and the mean in exponential distribution?
3. How does the exponential distribution model waiting times in real-world processes?
4. How can we estimate \( \lambda \) if we don't have complete data but know the average downtime?
5. Can the exponential distribution be used to model any type of data?

**Tip**: Exponential distributions are useful for modeling the time between independent events that happen at a constant average rate, such as machine failures or downtimes.

A manufacturing company monitors the downtimes on its key fabricating machines. Data for the downtimes on one particular machine are given in the accompanying data table. Assuming that the downtimes are exponentially distributed, use the data to find the probability that a specific downtime will exceed 0.9 minute.

This problem involves finding the probability that a machine downtime exceeds 0.9 minute, using the exponential distribution model. Learn how to apply the exponential distribution formula and estimate the rate parameter (λ) from the mean downtime. Suitable for undergraduate students studying probability theory and exponential distributions.

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